ALT In this paper, we present a geometric construction of contragredient Lie superalgebras, with the exceptions of A(m,n) (m,n>0) and G(3). Let 𝔤 be a complex simple Lie algebra with an order 4 automorphism σ, the eigenspace decomposition being 𝔤=𝔤(1)+𝔤(i)+𝔤(-1)+𝔤(-1). When 𝔤(i) admits a 𝔤(1)-invariant symplectic structure, (𝔤,σ) is called a generalized symmetric pair with symplectic subspace. In this paper, we classify all such pairs and construct a natural Lie superalgebra structure on L=L₀̄+L₁̄ with L₀̄=𝔤(1) and L₁̄=𝔤(i). Furthermore, we investigate the constants arising from the squares of two distinct Dirac operators: D for 𝔤=𝔤σ²+𝔤⁻σ², and Dₛ for L=𝔤(1)+𝔤(i). We prove that the constant C in D² admits a remarkable factorization C=J⁽ΩL_(₀̄))⋅dim L₁̄⋅ h∨_sign, where h∨_sign is the signed dual Coxeter number of L and J⁽ΩL_(₀̄)) is a mapped Casimir constant of its even part. This formula reveals an unexpected unification of symmetric space geometry and Lie superalgebra inva